If a and b are integers, and a is a factor of b, what must be true?

Question

If  a  and  b  are integers, and  a  is a factor of  b , what must be true?

I)  a < b

II) The distinct prime factors of  a^{2}  are also factors of  b .

III)  0<\frac{a}{b}\leq{1}

A) None
B) II only
C) III only
D) I and II only
E) II and III only

nick1816 · Answer

Give b=ka where a,b and k are integers
 
Statement 1- If k is -ve integer or equal to 0, a>b.
  a < b  can be true or false.

Statement 2- Distinct primes of a is same as that of  a^2 . Hence the distinct prime factors of  a^{2}  are also factors of  b .
Must be true

Statement 3- If b is negative multiple of a, then  \frac{a}{b}<0 
Hence  0<\frac{a}{b}\leq{1}  can be true or false

IMO B

LeoN88 · Answer

Must be true? Let's try to make them false-

b= ak
I 1 is a factor of -4 but 1>-4

II Always true, 4 is a factor of 48 then 2 (prime factor of 16) is also factor of 48

III Again 1 is a factor of -4 but 1/(-4) <0.

B) II only

ScottTargetTestPrep · Answer

Although a is a factor of b, a could equal b. So I is not true.

However, since the distinct prime factors of a^2 are also distinct prime factors of a, then they are also factors of b. So II is true.

If a is negative and b is positive, then a/b is negative. So III is not true.

Answer: B

aidyn · Answer

Hi, isn't factorial of negative number supposed to be undefined?

ScottTargetTestPrep · Answer

Yes, you’re right that the factorial of a negative number is undefined; however, this question does not involve any discussion about factorials. You might be confusing the term “factor” with “factorial,” which are very different things. A factor of an integer is another integer that divides the first integer without any remainder. For example, 2 and 3 are factors of 6, but 5 is not. Factorial, which is denoted by the exclamation mark (!), is the product of all the positive integers less than or equal to a given positive integer. For instance, 3! = 3 x 2 x 1, 5! = 5 x 4 x 3 x 2 x 1, etc.

aidyn · Answer

Understood. Thank you, Scott.

nguen · Answer

SO I understand the answer but need clarification. Should we always think of the possible negative factors? Or why would we in this case?

Because say if I’m asked for the factors of 6, i say it has 4 factors (only the +ve), either as a question or as an intermediary step in a question. There are many questions that test this logic without accounting for -ve factors

I’m really asking - what cue should we have to separate the 2 cases?

Is it bc a and b integers themselves can be + or -? So if the question starts by stating a and b are +ve integers - then we ignore the -be factors (like -3 as a factor of 6)

Posted from my mobile device

nguen · Answer

Although a is a factor of b, a could equal b. So I is not true.

However, since the distinct prime factors of a^2 are also distinct prime factors of a, then they are also factors of b. So II is true.

If a is negative and b is positive, then a/b is negative. So III is not true.

Answer: B

Hi, isn't factorial of negative number supposed to be undefined?

Yes, you’re right that the factorial of a negative number is undefined; however, this question does not involve any discussion about factorials. You might be confusing the term “factor” with “factorial,” which are very different things. A factor of an integer is another integer that divides the first integer without any remainder. For example, 2 and 3 are factors of 6, but 5 is not. Factorial, which is denoted by the exclamation mark (!), is the product of all the positive integers less than or equal to a given positive integer. For instance, 3! = 3 x 2 x 1, 5! = 5 x 4 x 3 x 2 x 1, etc.

ScottTargetTestPrep 
SO I understand the answer but need clarification. Should we always think of the possible negative factors? Or why would we in this case?

Because say if I’m asked for the factors of 6, i say it has 4 factors (only the +ve), either as a question or as an intermediary step in a question. There are many questions that test this logic without accounting for -ve factors

I’m really asking - what cue should we have to separate the 2 cases?

Is it bc a and b integers themselves can be + or -? So if the question starts by stating a and b are +ve integers - then we ignore the -be factors (like -3 as a factor of 6)

Posted from my mobile device

ScottTargetTestPrep · Answer

In a question involving factors of integers, it really depends on the context whether to include negative factors or not. Sometimes, the question will actually tell you to consider only the positive factors while other times, a negative factor will not be applicable if it stands for the number of people in a room or the distance traveled by a car etc. If the question mentions nothing about the sign of the factors and if negative factors make sense to consider, such as the case for this question, you should not disregard negative factors.

ManifestDreamMBA · Answer

Can someone please help clarify? I am wondering about statement II, with a= 1, where a has no prime factor. Does that mean a= 1 is not a valid case? Or we can still use it and think of this as true given there are no prime factors of 1

If a and b are integers, and a is a factor of b, what must be true?

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If a and b are integers, and a is a factor of b, what must be true?

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